Photons are massless, therefore, in the Standard Model, the Higgs boson does not couple directly to photons. Instead, the decay of the Higgs into 2 photons is a loop mediated process. It happens to be dominated by two contributions: one, larger, from the W boson, and the other, 5 times smaller, from the top quark. Incidentally, these two contributions enter with opposite signs (in fact it's not quite an accident but a consequence of the theorem that links these contributions to the electromagnetic beta function). The other known charged particles are expected to contribute much less because their coupling to the Higgs is much smaller.
Now it is clear what tricks can be played to pump up the Higgs decay width into photons:
- Increase the Higgs couplings to W bosons. But then one needs to explain why no excess in the WW channel has been observed.
- Decrease the Higgs coupling to the top quark. But then the Higgs production rate would be decreased as well, and one needs to cook up new production channels.
- Introduce one (or more) new charged particle that contributes as much as the top quark, but with the opposite sign.